Perfect Numbers

A perfect number is a whole number which is equal to the sum of its divisors including 1 by excluding the number itself. Thus 6 is a perfect number becauses 1 + 2 + 3 = 6. Likewise 28 is a perfect number because 1 + 2 + 3 + 4 + 7 + 14 = 28.

Leonard Euler (1707-1783), a mathematician born in Switzerland but who worked in Germany and Russia, proved that every even perfect number is of the form 2^p-1 (2p -1) where p is a prime and 2^p -1 is also a prime, called a Mersenne prime in honor of Mersenne (1588-1648), a Franciscan friar who often served as an intermediary in the correspondence between the most prominent mathematicians of his time. For example, if p = 2, a prime, then 2^{p -1} = 2² -1 = 4-1 = 3 is also a prime and 2 ^p-1(2^p -1) = 2 × 3 = 6 which, as we have seen is indeed a perfect number. If p = 3, the next prime, then 2^p -1 = 2³ 1 = 8-1 = 7 is again a prime. This gives us 2² × 7 = 4 × 7 = 28 which, as we have seen, is also a perfect number. The first case where p is a prime but 2^p -1 is not occurs when p = 11. Here we have 2¹¹ -1 = 2048-1 = 23 × 89.

The search for even perfect numbers, then, is the same as the search for Mersenne primes. At the present time 32 are known corresponding to p = 2, 3, 7, 13,..., 756,839 where the last one, as well as some of the others, were found by computers. The largest of these, 756,839 yields a perfect number of 227,832 digits.

At the present time no odd perfect numbers are known. It is suspected that none exist and this hypothesis has been tested by computers up to 10³⁰⁰ but, of course, this does not constitute a proof that none exist.

The terminology of "perfect" goes back to the Greeks of Euclids time who would personify numbers. Thus if the sum of the divisors was less than the number, the number was called deficient. If the sum was greater than the number it was called abundant.